Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Answer the following questions (Any seven only):
Show that $ f(x)=\sin^{2}x $ is continuous for every value of x.
Verify Rolle's Theorem for the function $ f(x)=2+(x-1)^{2/3} $ where $ x\in[0,2] $.
Test the convergence of the integral $ \int_{0}^{\infty}\frac{\cos x}{1+x^{2}}dx $.
Evaluate $ \Gamma(-\frac{5}{2}) $.
Define Power series and Taylor's series.
Find the smallest positive period of the function $ \sin(\frac{n\pi x}{L}) $.
Are the following set of vectors $ [1, -1, 1], [1, 1, -1], [0, 1, 0] $ linearly independent or dependent?
Define rank and nullity of a matrix.
Find the spectral radius of the matrix $ A=\begin{bmatrix}0&1\\ -1&0\end{bmatrix} $.
Show that the determinant of an orthogonal matrix has the value +1 or -1.
Q.2 Solve both questions :
Verify Lagrange's mean value theorem for $ f(x)=(x-1)(x-2)(x-3) $ in $ [0,4] $.
Find the maxima and minima values of the function $ f(x)=5x^{6}+18x^{5}+15x^{4}-10 $.
Q.3 Solve both questions :
Find the evolute of the parabola $ y^{2}=4ax $.
Evaluate the integral $ \int_{0}^{\pi/2}\sqrt{\tan \theta}d\theta $.
Q.4 Solve both questions :
Find the radius of convergence of the series $ \sum_{m=0}^{\infty}(m+1)m~x^{m} $.
Expand the function $ f(x)=2x^{3}+7x^{2}+x-6 $ in powers of $ (x-2) $.
Q.5 Solve both questions :
Find the half range sine Fourier series of the periodic function $ f(x)=\pi-x, \quad 0 < x < \pi $.
Solve the system of equations by Cramer's rule:
$ -x_{1}+x_{2}+2x_{3}=2 $
$ 3x_{1}-x_{2}+x_{3}=6 $
$ -x_{1}+3x_{2}+4x_{3} =4 $
Q.6 Solve both questions :
Find the inverse of the matrix $ \begin{bmatrix}3&-1&5\\ 2&6&4\\ 5&5&9\end{bmatrix} $ by Gauss-Jordan Elimination.
Can we say all vectors in $ R^{3} $ such that $ 4v_{2}+v_{3}=k $ form a vector space? If yes, determine the dimension and find a basis.
Q.7 Solve both questions :
Find the nullity of the matrix $ \begin{bmatrix}4&0&2&8\\ 5&7&3&1\\ 0&6&9&0\end{bmatrix} $.
Find a basis of eigenvectors of the matrix $ A=\begin{bmatrix}2&1\\ 2&1\end{bmatrix} $.
Q.8 Solve both questions :
Find the eigenvalues and eigenvectors of the matrix $ A=\begin{bmatrix}5&4\\ 1&2\end{bmatrix} $.
Diagonalize the matrix $ A=\begin{bmatrix}-2&2&-3\\ 2&1&-6\\ -1&-2&0\end{bmatrix} $.
Q.9 Write short notes on any two of the following:
Mean value theorem
Beta Function
Vector Space
Gauss Elimination
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Answer the following questions (Any seven only):
Show that $ f(x)=\sin^{2}x $ is continuous for every value of x.
Verify Rolle's Theorem for the function $ f(x)=2+(x-1)^{2/3} $ where $ x\in[0,2] $.
Test the convergence of the integral $ \int_{0}^{\infty}\frac{\cos x}{1+x^{2}}dx $.
Evaluate $ \Gamma(-\frac{5}{2}) $.
Define Power series and Taylor's series.
Find the smallest positive period of the function $ \sin(\frac{n\pi x}{L}) $.
Are the following set of vectors $ [1, -1, 1], [1, 1, -1], [0, 1, 0] $ linearly independent or dependent?
Define rank and nullity of a matrix.
Find the spectral radius of the matrix $ A=\begin{bmatrix}0&1\\ -1&0\end{bmatrix} $.
Show that the determinant of an orthogonal matrix has the value +1 or -1.
Q.2 Solve both questions :
Verify Lagrange's mean value theorem for $ f(x)=(x-1)(x-2)(x-3) $ in $ [0,4] $.
Find the maxima and minima values of the function $ f(x)=5x^{6}+18x^{5}+15x^{4}-10 $.
Q.3 Solve both questions :
Find the evolute of the parabola $ y^{2}=4ax $.
Evaluate the integral $ \int_{0}^{\pi/2}\sqrt{\tan \theta}d\theta $.
Q.4 Solve both questions :
Find the radius of convergence of the series $ \sum_{m=0}^{\infty}(m+1)m~x^{m} $.
Expand the function $ f(x)=2x^{3}+7x^{2}+x-6 $ in powers of $ (x-2) $.
Q.5 Solve both questions :
Find the half range sine Fourier series of the periodic function $ f(x)=\pi-x, \quad 0 < x < \pi $.
Solve the system of equations by Cramer's rule:
$ -x_{1}+x_{2}+2x_{3}=2 $
$ 3x_{1}-x_{2}+x_{3}=6 $
$ -x_{1}+3x_{2}+4x_{3} =4 $
Q.6 Solve both questions :
Find the inverse of the matrix $ \begin{bmatrix}3&-1&5\\ 2&6&4\\ 5&5&9\end{bmatrix} $ by Gauss-Jordan Elimination.
Can we say all vectors in $ R^{3} $ such that $ 4v_{2}+v_{3}=k $ form a vector space? If yes, determine the dimension and find a basis.
Q.7 Solve both questions :
Find the nullity of the matrix $ \begin{bmatrix}4&0&2&8\\ 5&7&3&1\\ 0&6&9&0\end{bmatrix} $.
Find a basis of eigenvectors of the matrix $ A=\begin{bmatrix}2&1\\ 2&1\end{bmatrix} $.
Q.8 Solve both questions :
Find the eigenvalues and eigenvectors of the matrix $ A=\begin{bmatrix}5&4\\ 1&2\end{bmatrix} $.
Diagonalize the matrix $ A=\begin{bmatrix}-2&2&-3\\ 2&1&-6\\ -1&-2&0\end{bmatrix} $.
Q.9 Write short notes on any two of the following:
Mean value theorem
Beta Function
Vector Space
Gauss Elimination